# How can the magnetic field surrounding a current-carrying wire ever be uniform?

My book says that to find the force a current-carrying wire exerts on a moving charge, one uses $B = (\mu_0 / 2\pi)(I/r)$ to find the magnitude of magnetic field around the wire, and then uses that to find $F_B = q v B \sin\theta$, which is the same formula as that for uniform field between the planes of a permanent magnet.

How can the field around the wire ever be uniform since intuitively it weakens as distance increases?

• Two possibilities here, one you are looking at a question of example that says "Assume a wire runs through a region of uniform magnetic field.." in which case it is not talking about the field from the wire but about some externally imposed field. – dmckee --- ex-moderator kitten Jul 28 '15 at 22:51
• Secondly, when we want to make a uniform field we don't use a single, long, straight wire to do it. It won't be long before you are shown geometries that can achieve that to a very good approximation indeed. – dmckee --- ex-moderator kitten Jul 28 '15 at 22:52
• @dmckee, no, it's impossible. It's really elementary, won't be that complex – most venerable sir Jul 28 '15 at 22:52
• @dmckee, in case i getting it wrong: " the equation for the force of magnetism acting on the charge through a current carrying Wire is the same as when a charge moves in a uniform magnetic field." – most venerable sir Jul 28 '15 at 22:57
• Please don't post photos of books, diagrams, or anything else that you can type into the question body. – DanielSank Jul 29 '15 at 2:17

The field around the wire isn't uniform. When you calculate the force on a charge in a magnetic field, you use the value of the field at the point where the particle is. So, $$F_B = qvB\sin\theta$$ is not just for a constant field. If the field varies from position to position in space, then the force the particle feels will also vary. This equation is introduced with a constant field because the math is easier and because the resulting motion is a simple circle.
So, when you calculate the force of a current-carrying wire on a particle, the resulting force is only for that moment in time. If the particle moves towards or away from the wire, the force on the particle will be different. $F_B = qvB\sin\theta$ is always true, but that doesn't mean the force is constant.
Below is a picture of a proton traveling at $10\ m/s$ next to a wire carrying $1\ A$ of current. The proton starts off at the bottom of the picture traveling upwards parallel to the wire in the same direction as the current. Notice that as the proton gets closer to the wire, the magnetic field increases, so the turn gets tighter. Once the proton turns $180^o$, it starts moving away from the wire, and the turns become wider.